I think you are over dramatizing, and over complicating the situation. People have been keeping fish and plants successfully for hundreds of years. It's not rocket science.
Your plan needs to revolve around what you want. By the sounds of it you want a low tech setup with slow growth and minimal attendance to the tank. This is the easiest setup. Why over complicate the situation with adding soil?
Grab a decent low light for the tank. A finnex fuger ray would be fine. If you don't want to spend a lot of money then use a 9 watt 5000k flood light (not the narrow beam spot light) LED from home depot or lowes
Get some fluorite or eco complete (about 20 lbs for a 10 gal). Wash it well. Put it in the tank. Connect all the filtration and heating equipment. Wait a day or so to settle. Put in root tabs. Add hearty plants like anubias, swords and val. Watch them grow.
Add your choice of fish slowly. 3 or 4 hearty zebra danios will cycle your tank fine without any deaths if you feed 1x a day...just a pinch. You can help the cycling process if you have a friend with an established tank (just a generous handful will suffice). You can take some of his gravel and it put it in the tank after you add fish.
For the first week after you add fish change 25% (add prime for NYC water) of the water every other day. Do the same for the second week if you can.
If you notice signs of stress in the fish change 50% of the water.
After 6 weeks the tank will practically take care of itself. Do monthly 50% water changes.
That's all there is to it.
Who knows...after you have success with this method you might want to upgrade to more challenging species with co2 injection and higher light.
To answer your questions directly:
What I originally wanted was a lush, green 10 gallon tank with a few neon tetras (my favorite since childhood). Is this not reasonable? I've heard that small tanks are hard.
Small tanks are no harder than large tanks. As long as it has a filtration unit and a stable heater most things are possible. Neon tetras are a good choice, but I would stay away from them for the first few months. They don't handle ammonia and nitrite toxicity well. It's important to choose a beginning fish according to what they are used to in nature. For all practical purposes a beta is basically indestructible against ammonia and is the best choice for survival, but it may not be what you want in the tank in the long run (a potentially aggressive fish...females are not, though).
Having a green, lush tank is possible. In a low light setup with no co2 plants will not grow fast, but under optimal conditions will grow healthy. I'm not sure how NYC water is for growing plants, but I've heard it's great. There should be plenty of micro nutrients in the tank for plants to thrive. If you run into yellow, decaying growth down the road be sure to visit TPT again and we can suggest a comprehensive fertilizer you can use to ensure success in a low light, slow growing tank.
Is a planted tank too complicated and should I just opt for just a regular tank with plastic plants, clown puke gravel, a treasure chest aerator that bobs open and closed, and fish that die every 8 weeks?
The plants...in the long run atleast will be easier to maintain than your fish. As stated above you MIGHT run into a situation where you need to add a half capful of seachem's flourish once or twice every few weeks. You will need to replace root tabs for the plants approx. every 2-3 months. I suggest seachem's root tabs because they are complete with trace elements.
I would stay away from the chest that bobs open and closed with fish dying every few weeks
A word of warning: Fish can die. They die in nature of disease, predation and natural causes. All of these things can happen in your aquarium as well. Don't let the death of fish discourage you. Find the root cause. Usually fish die of disease following an acute toxicity of ammonia or nitrite during the first few months of setting up a tank. Diseases can happen anytime, but if the plants are healthy and keeping the water clean the fish will have a bolstered immune system and shouldn't succumb as easily.
Algae: Don't let this discourage you. It will probably happen in the first few months. Green water, brown gunk on plant's leaves, and on the substrate. Less likely to be a huge problem in a low light tank, but chances are good you will have atleast some algae from time to time; especially during the first few months.
Is the nano tank with shrimp and snails a good approach for beginners
I would stick with the 10 gal plan with a few hearty fish and plants to begin with. Shrimp depending on the type can be a chore to keep...especially if you plan on keeping fish with them. Snails are always good and I love them for eating algae. Don't get any snails that reproduce like crazy.
I'd say good luck, but it's not needed. You will succeed.